Bound on Entanglement in Neural Quantum States

Area of Science:

• Quantum Physics
• Computational Physics

Background:

• Variational wave functions provide computational advantages for many-body systems but often face limitations in expressive power.
• Matrix product states are efficient for certain quantum states but are restricted to systems obeying an area law for entanglement.

Purpose of the Study:

• To investigate the fundamental constraints on the expressive power of neural quantum states (NQS).
• To establish a theoretical bound on the entanglement entropy for feed-forward neural quantum states.

Main Methods:

• Analysis of feed-forward neural quantum states with n spins and k scalar nonlinearities.
• Application of analyticity assumptions to derive entanglement entropy bounds.
• Analytical and numerical demonstrations of the tightness of the derived scaling.

Main Results:

• A fundamental constraint on entanglement entropy for NQS is proven: S≤cklog n.
• This result establishes an analog of the area law constraint for NQS.
• Volume law entanglement is ruled out for NQS with a fixed number of nonlinearities.
• The derived scaling with n is shown to be tight for various NQS architectures.

Conclusions:

• Neural quantum states, despite their potential, are subject to fundamental constraints on entanglement.
• The established bound S≤cklog n clarifies the limitations of NQS, particularly concerning volume law entanglement.
• This work provides a crucial understanding of NQS capabilities and limitations across diverse network designs.